{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

hw7 - 72 cHAprER z THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 72 cHAprER z. THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN 7 . 1 1 o [ n e w ] ( a ) z z * : ( x + i y ) ( r - i A ) = 1 2 I y 2 = l z l 2 . (b) We first note that, as you can easily check, (zw)* : z*ru*. Then, lzwl2 = zw(2y1)" : zwz*w* : (zz-)(uw-) : lzl2lul2. Therefore lzwl: lzl.lul. (c) If v(r, t) : {(x)e-tut, then lv(r,t)l= l{(')t.le--tl : lr/(c)1. 7.L2 o [old,8.t0] (a) z =re-tut: r(cos &rt-i sin wt) = a-60, wherez: rcos(trt) andy:asi11r1;. 7.13 or [old 8.tt] (a) If. g : asst1r1l * bsin(c,.rl), we can define A: 1;T+F, so that ,4 is the hypotenuse of a triangle with sides a, b, and .4. If we call / the angle opposite a, lQ = arcstn(a/A)l then sin / : alA and, cos d : b/?4. T'hus a = Al(a/A) cos at + (blA)sinutl = -A[sin dcosut + cos /sint.rt] : Asin(tut + d). Conversely, if y -- Asin(ut ! d) : ,4 sin dcos t,,'t t Acos$sinult, then we can define a: Asind and b: Acosd and then y : acosut * bsinut, (b) If g: Asin(ut + Q), we let t' =t+flu and then y: Asin(o-rt'). 7.L4 co [old,8.r2] eie : 7 t Uq + Uq'z Pt + (iq3 ftt + ... : t + i0 - 02 lzt - i03 lsl + 04 l4t + i02 lst -... : G - 02 /2t + 04 l4l -...) + i(0 - es /sl + 05 lil - ...) : cos I * isind. 7.15 oo [new] F]om Euler's relation, we know that ei(e+o) : cos(9 + {) + i sin(d + {). (I) Also, by the well known property of exponentials, ei(0+6):eieeid:(cosl+isint)(cosd+isin/):(cosgcos/-sindsin{)+i(sin0cos/+cosdsin/). Comparing this with Eq, (I) above and equating real and imaginary parts, we find that cos(0 + S) : cos , cos d - sin I sin / and sin(9 + /) : sin d cos d * cos d sin / 7.16 oo fold 8.13] FYom E,q. (7.121) we know that8....
View Full Document

{[ snackBarMessage ]}

Page1 / 4

hw7 - 72 cHAprER z THE SCHRODNGER EQUATIoN rN oNE DIMENSIoN...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online